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Comment from S. Pollock (CU Boulder, visiting OSU and teaching Paradigm “Vector Fields), Nov 2009:

I started this activity at the end of lecture #1, but only gave them about 6 minutes for it. I specified E and the cone (simplified it to a 45 degree cone of height 1) and asked for the outward flux on the “cone-surface” (not the top).

In this short time, all groups had sketched the E field, and started thinking about appropriate dA's to use. Most groups spent some time thinking about choice of coordinate system (most picked cylindrical, one used spherical [which would work!] and one was trying Cartesian, but ended up realizing this was hard). None spontaneously used the formal method I will lead to (of constructing the dA vector by crossing dr1 x dr2, where dr1 and dr2 can each be written of the general form, say in Cylindrical coords, of dz zhat + dr rhat + r dphi phihat, and then “using what you know about the path” to simplify/re-express in terms of simple integration variable.s) Instead, they were just trying to look at the areas geometrically and craft an expression - which isn't wrong, and in fact works just fine. I let this go, but then we moved on to another activity.

I followed this up at the start of next class (lecture 2) by talking through the dr story (above) on the board, and then let them have at it again ( 5 minutes of lecture on this). This time they zoomed in more quickly on productive approaches, all groups tackled the issue of re-writing the “z” in the E-field in terms of the variables they were integrating over (r) I gave them about 12 minutes more, at which point 2/6 groups had just finished (with correct solution, including sign) I then completed the problem for them (with guidance and frequent questions) in cylindrical, and then sketched what would be the same/different if you used spherical.

I had anticipated this would be a formidably hard activity, but because of the simplifications I made (simple cone, simple geometry, especially in spherical coordinates) they seemed to access the key points (construction of dA, setting up the integration limits, taking the dot product, converting variables based on the particular surface) in a reasonable time (less than half an hour all together?)

Thoughts: if you WANT students to be able to set up and evaluate non-trivial surface integrals, this seems like an excellent activity. I am not sure this is a skill that I care so much about for my non-Paradigms course (it's not emphasized once we leave Ch1 of Griffiths!) so one must consider carefully what the course/learning goals are. But in terms of helping students transfer their (weak, fragmented in my case) math class vector calculus skills to this particular E&M context, it worked.

-S